Stokes Matrices and Monodromy of the Quantum Cohomology of Projective Spaces

In this paper we compute Stokes matrices and monodromy of the quantum cohomology of projective spaces. This problem can be formulated in a “classical” framework, as the problem of computation of Stokes matrices and monodromy of differential equations with regular and irregular singularities. We prove that the Stokes' matrix of the quantum cohomology coincides with the Gram matrix in the theory of derived categories of coherent sheaves. We also study the monodromy group of the quantum cohomology and we show that it is related to hyperbolic triangular groups. Received: 24 October 1998 / Accepted: 27 April 1999     

Global wellposedness for quasi-linear symmetric hyperbolic systems with regularized coefficients

Following the abstract setting of [8] and using the global results of [2], global wellposedness and regularity results are proved for the solutions of quasi-linear symmetric hyperbolic systems with bounded coefficients which are regularized by a convolution in the space variables with a regularizing function. In the case of unbounded regularized coefficients, local existence of classical solutions is proved, as well as uniqueness and regularity of (not necessarily existing) global weak solutions with initial value in a Sobolev space. As the regularizing function tends to Dirac's δ, local-in-time convergence to the classical solution of the non-regularized problem is proved.     

Symmetry Groups and Non-Planar Collisionless Action-Minimizing Solutions of the Three-Body Problem in Three-Dimensional Space

Periodic and quasi-periodic solutions of the n-body problem can be found as minimizers of the Lagrangian action functional restricted to suitable spaces of symmetric paths. The main purpose of this paper is to develop a systematic approach to the equivariant minimization for the three-body problem in three-dimensional space. First we give a finite complete list of symmetry groups fitting to the minimization of the action, with the property that any other symmetry group can be reduced to be isomorphic to one of these representatives. A second step is to prove that the resulting (local and global) symmetric action-minimizers are always collisionless (when they are not already bound to collisions). Furthermore, we prove some results which address the question of whether minimizers are planar or non-planar; as a consequence of our theory we will give general criteria for a symmetry group to yield planar or homographic minimizers (either homographic or not, as in the Chenciner-Montgomery eight solution). On the other hand we will provide a rigorous proof of the existence of some interesting one-parameter families of periodic and quasi-periodic non-planar orbits. These include the choreographic Marchal's P₁₂ family with equal masses – together with a less-symmetric choreographic family (which anyway probably coincides with the P₁₂ family).     

Sol‐to‐Gel Transition in Fast Evaporating Systems Observed by in Situ Time‐Resolved Infrared Spectroscopy

The in situ observation of a sol‐to‐gel transition in fast evaporating systems is a challenging task and the lack of a suitable experimental design, which includes the chemistry and the analytical method, has limited the observations. We synthesise an acidic sol, employing only tetraethylorthosilicate, SiCl₄ as catalyst and deuterated water; the absence of water added to the sol allows us to follow the absorption from the external environment and the evaporation of deuterated water. The time‐resolved data, obtained by attenuated total reflection infrared spectroscopy on an evaporating droplet, enables us to identify four different stages during evaporation. They are linked to specific hydrolysis and condensation rates that affect the uptake of water from external environment. The second stage is characterized by a decrease in hydroxyl content, a fast rise of condensation rate and an almost stationary absorption of water. This stage has been associated with the sol‐to‐gel transition.     

Sub-Riemannian Curvature in Contact Geometry

We compare different notions of curvature on contact sub-Riemannian manifolds. In particular, we introduce canonical curvatures as the coefficients of the sub-Riemannian Jacobi equation. The main result is that all these coefficients are encoded in the asymptotic expansion of the horizontal derivatives of the sub-Riemannian distance. We explicitly compute their expressions in terms of the standard tensors of contact geometry. As an application of these results, we obtain a sub-Riemannian version of the Bonnet–Myers theorem that applies to any contact manifold.     

Holderian Weak Invariance Principle for Stationary Mixing Sequences

We provide some sufficient mixing conditions on a strictly stationary sequence in order to guarantee the weak invariance principle in Hölder spaces. Strong mixing and \(\rho \)-mixing conditions are investigated as well as \(\tau \)-dependent sequences. The main tools are deviation inequalities for mixing sequences.     

Uniaxial Modeling of Multivariant Shape-Memory Materials with Internal Sublooping using Dissipation Functions

Models for the macroscopic behavior of Shape Memory Materials can be conveniently constructed within the Ziegler–Green–Naghdi framework where all the constitutive information is encoded in two ingredients: the free energy and the dissipation function. In a previous work, we have proposed various expressions for the basic functions suitable to model pseudoelasticity with complete transformations cycles. In this work we consider additional effects due to Martensite reorientation and to transformation reversal prior to transformation completion. The new constitutive model allows for the modeling of a variety of effects including: shape memory associated with thermally induced transformation, internal pseudoelastic subloops and the determination of limit cycles associated with repetitive stress cycling.     

Models for one-variant shape memory materials based on dissipation functions

Some simple models for the macroscopic behavior of shape memory materials whose microstructure can be described as a mixture of two phases are derived on the basis of a free energy and a dissipation function. Keeping a common expression for the free energy, each model is based on a different expression for the dissipation function. Temperature-induced as well as isothermal, adiabatic and convective stress-induced transformations are studied. Attention is paid to closed form solutions, comparison among the models and parameter identification.